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Questions and Answers

What is the expansion of sin(x) in terms of x?

  • x + x^2 + x^3 +...
  • x^2 - x^4/3! + x^6/5! -...
  • x - x^3/3! + x^5/5! -... (correct)
  • 1 - x^2/2! + x^4/4! -...
  • What is the limit of f(x) as x approaches a, if f(x) is an algebraic function?

  • It is a logarithmic limit
  • It is a trigonometric limit
  • It is a exponential limit
  • It is a algebraic limit (correct)
  • What is the expansion of cos(x) in terms of x?

  • 1 + x + x^2/2! + x^3/3! +...
  • 1 - x^2/2! + x^4/4! - x^6/6! +... (correct)
  • x^2 - x^4/3! + x^6/5! -...
  • x - x^3/3! + x^5/5! -...
  • What is the expansion of tan(x) in terms of x?

    <p>x + x^3/3! + x^5/5! +...</p> Signup and view all the answers

    What is the expansion of e^x in terms of x?

    <p>1 + x + x^2/2! + x^3/3! +...</p> Signup and view all the answers

    What is the value of lim x → 0 (sin 2x / x)?

    <p>2</p> Signup and view all the answers

    If f(x) = 2 sin(x) and g(x) = 3 cos(x), then what is the value of lim x → 0 (f(x) / g(x))?

    <p>2/3</p> Signup and view all the answers

    What is the derivative of the function f(x) = tan(2x) using the chain rule?

    <p>2 sec^2(2x)</p> Signup and view all the answers

    What is the value of lim x → 1 (G(x) - G(1)) / (x - 1) if G(x) = 25 - x^2?

    <p>1/24</p> Signup and view all the answers

    What is the value of lim x → a (g(x) f(a) - g(a) f(x)) / (x - a) if f(a) = 2, f'(a) = 1, g(a) = 1, and g'(a) = 2?

    <p>3</p> Signup and view all the answers

    What is the value of lim(x→∞) ((x+1)^2)/(x^2+2x)?

    <p>1</p> Signup and view all the answers

    If a, b, c, and d are positive, then what is the value of lim(x→∞) ((a+bx)/(c+dx))?

    <p>e^(d/b)</p> Signup and view all the answers

    What is the value of lim(x→∞) (1+x)^(1/x)?

    <p>e</p> Signup and view all the answers

    What is the value of lim(x→0) (sin(x)/x)?

    <p>1</p> Signup and view all the answers

    What is the value of lim(x→∞) (x^2+x-1)/(x^2+2x)?

    <p>1</p> Signup and view all the answers

    What is the value of lim h → 0 (1 - cos(2h)) / sinh?

    <p>1/2</p> Signup and view all the answers

    What is the value of lim x → 0 (x^2 sin(3x)) / sin(5x)?)

    <p>0</p> Signup and view all the answers

    What is the value of lim x → 0 (tan(x) - sin(x)) / x^2?)

    <p>1/6</p> Signup and view all the answers

    What is the value of lim x → 0 (sin(2x) / sin(5x))?)

    <p>2/5</p> Signup and view all the answers

    What is the value of lim x → 1 (f(x) - f(1)) / (x - 1) if f(x) = 2 sin(x)?)

    <p>2</p> Signup and view all the answers

    What is the value of lim x → 0 (sin(x) / x^2)?)

    <p>1/3</p> Signup and view all the answers

    What is the value of lim x → 0 (cos(2x) - 1) / sin(x)?)

    <p>-2</p> Signup and view all the answers

    What is the value of lim x → 0 (x^3 sin(3x)) / sin(5x)?)

    <p>0</p> Signup and view all the answers

    What is the value of lim x → 0 (sin(3x) / sin(5x))?)

    <p>3/5</p> Signup and view all the answers

    What is the value of lim x → 0 (tan(2x) - sin(2x)) / x^2?)

    <p>1/6</p> Signup and view all the answers

    What is the Taylor series expansion of the function log(1 + x) around x = 0?

    <p>x - x^2/2 + x^3/3 -...</p> Signup and view all the answers

    What is the condition for the function f(x) = (1 + 1/x)^x to be continuous at x = 0?

    <p>The function is not continuous at x = 0.</p> Signup and view all the answers

    What is the derivative of the function f(x) = sin^(-1)x using the chain rule?

    <p>1 / sqrt(1 - x^2)</p> Signup and view all the answers

    What is the limit of the function f(x) = (1 + x)^1/x as x approaches 0?

    <p>e</p> Signup and view all the answers

    What is the Taylor series expansion of the function cosh(x) around x = 0?

    <p>1 + x^2/2 + x^4/4! +...</p> Signup and view all the answers

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